"There's nothing in here to change the consensus assumption regarding rate hikes, which center on the middle of next year," said Daniel Greenhaus, chief global strategist at BTIG.
"They're clearly in a position where at least right now, they're inclined to let this thing run a little further, to take out some insurance. Given how long we have under performed, if you're the Fed and you view things the way they do, what's the harm in going a little further?"
Market focus now shifts to Fed Chair Janet Yellen's testimony before Congress next week, and the comments of other Fed officials on the speaking circuit.
Read MoreFed minutes: QE likely to end with $15 billion in October
On Thursday, Fed Vice Chairman Stanley Fischer will speak at 4:30 p.m. (EST) on financial sector reform at a conference hosted by the National Bureau of Economic Research. There are also jobless claims data at 8:30 a.m. and wholesale trade at 10 a.m.
The bond market has been beginning to wager, particularly after that jobs report, that the Fed would move sooner than it expects to hike rates. The two-year yield reached a three-year intraday high of 0.53 percent, while the long end held in its recent trading range.
This so-called "curve-flattening trade" was viewed a sign the market was sensing the Fed may have to become less accommodative because of an improving economy. Boockvar noted the yield curve remained flat even with the big move lower in yields post-Fed minutes.
Jeff Rosenberg, chief investment officer of fixed income at BlackRock, said the two-year yield had been moving on the idea the economy is improving enough to force the Fed to hike rates.
"The data is showing evidence that they're closer to the attainment of their dual objectives," he said earlier Wednesday. The two-year yield backtracked to 0.48 percent after the Fed release.
Rosenberg said he believes the Fed should move to raise its target Fed funds rate from zero by the first quarter. The Fed has had a soft target of 6 percent unemployment and a 2 percent inflation rate.
Read MoreWill the Fed fuel more volatility?
Fixed income experts expect to see a continued tug of war in the market, reflecting the diverging views on Fed interest rate policy, particularly if economic data improves.
"I think this will contribute to an increase in volatility, all else being equal. There's a divergence in opinions around the Fed exit, and the various implications of that will increase volume. To what degree, I don't know," said Gregory Peters, portfolio manager at Prudential Financial.
The June jobs report added juice to the recent selloff of the short end of the curve, as traders viewed the solid employment gain of 288,000 and unemployment rate of 6.1 percent as a sign the labor market is on the mend. However, the number of long-term unemployed remains high, and the 63.8 percent participation rate is still at a post-recession low.
"The concern is the short-term unemployment rate is the better measure of slack, and policy is too accommodative," Rosenberg said. That could lead to concern about financial complacency and financial stability longer term if the Fed stays on hold for too long, he said.
In the minutes, it noted that Fed officials discussed "whether some recent trends in financial markets might suggest that investors were not appropriately taking account of risks in their investment decisions."
A changing forecast
Following the June jobs report, some economists moved forward their expectations for when the Fed will start raising rates. Goldman Sachs economists, for example, changed their forecast from the first quarter of 2016 to the third quarter of 2015.
J.P. Morgan economists also expect a third-quarter, 2015 rate hike. In a note Wednesday, they said the gap between lowering the short- versus the long-term unemployment rate is quickly narrowing.
"The debate as to whether short-term unemployment or total unemployment is more important from an inflation perspective really heated up at the beginning of the year. At that point the disconnect between short-term and long-term unemployment was quite wide," they noted.
The economists said since then, the divergence has narrowed with the short-term unemployment rate at 4.1 percent in June from 4.2 percent in December but a market larger drop from 2.5 percent in the long-term rate in December to 2 percent in June.
They also noted that long-term joblessness, while still elevated, accounted for more than 90 percent of the decline in unemployment since December.
The Treasury auctions $13 billion of reopened 30-year bonds Thursday after Wednesday's auction of $21 billion in 10-year notes Wednesday afternoon was met with a tepid response. The yield of 2.59 percent was the lowest in three years.
Ian Lyngen, senior Treasury strategist at CRT Capital, said the auction was weak, as expected as investors anticipated the Fed minutes, released an hour later. But after the Fed minutes, the 10-year yield fell back to 2.55 percent.
Fed funds futures also moved Wednesday, and after the Fed minutes, rates were slightly lower. They imply a rate of 0.55 percent by September, 2015 and 0.78 percent by December, 2015.
The median forecast of Fed officials is that rates will be 1.2 by the end of next year and 2.5 percent by the end of 2016.
"I think that's going to be the tension in here. I think as the data come in a little stronger and inflation comes in just a little higher, I think the markets are going to be just a little more skittish, and I think that's what you're seeing," Peters said. "I do not personally believe they're behind the curve. But that's the tension in the market and that's going to be a source of volatility in the market. It's showing up in the front end."
The real barometer
Peters said the issue isn't really when the Fed starts raising rates, but more so the pace of change and what the rate will be by the end of next year.
"From an investment standpoint, it matters more where it ends up. It's a time when investors have become accustomed to a highly accommodative policy, so a move in the opposite direction matters," Peters said. "What history tells you is once they move, they continue to move, so it's a progression. I think the hair-trigger response from the market is one thing, but from an investment standpoint that's infinitely more important."
Peters said the price action makes sense from the perspective that the long end of the curve is attractive when compared to other sovereigns in Europe and Asia. He said the Fed has already been tightening as it tapers its quantitative easing bond buying, but he doesn't believe the market will react the way it has in the past to tapering. Peters also points out the action this time is in the front end of the curve.
"Either way, you're getting closer to the Fed doing something than further away. I do think the Fed and (Fed Chair Janet) Yellen believe inflation isn't a problem. They want to get growth up. There's still slack," he said, adding the Fed will stress that rates stay lower for longer.
"Ultimately, the market's really going to push on that and the Fed's job is going to be a little more difficult here for the next couple months, given the rebound from the first quarter and the weaker inflation prints rolling off—and the market's going to test," said Peters.
While the CPI has shown consumer inflation running at 2 percent year over year, the Fed's preferred measure, personal consumption expenditures, is still lagging.